my boolean modifier isn't cutting into my shape correctly
I downloaded your file and the operations worked as follows:


Apply the mirror modifier on object B, Cube in the Outliner  
Make sure only object B is selected, which should be because you just applied the mirror modifier
Add a Boolean modifier and select for the Operation the Difference, see image below with the mark 1
Click on the eye dropper, mark 2  
Select object A, mark 3. The object name Cube.001 in the Outliner.





Hide object A, Cube.001, from the Viewport and from the Render, see image below. The results should be visible without applying the operation but you can apply it at this point if you wish.